Cumulative Assessment
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Recall the classification of quadrilaterals. You can begin by finding the slopes of the sides.
See solution.
We will start by plotting the given points on a coordinate plane and graphing the quadrilateral.
| Quadrilateral | Definition |
|---|---|
| Parallelogram | Both pairs of opposite sides are parallel. |
| Rhombus | A parallelogram with four congruent sides. |
| Rectangle | A parallelogram with four right angles. |
| Square | A parallelogram with four congruent sides and four right angles. |
| Trapezoid | A quadrilateral with exactly one pair of parallel sides. |
| Isosceles Trapezoid | A trapezoid with legs that are congruent. |
| Kite | A quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent. |
To prove that quadrilateral WXYZ is a rectangle we need to show that both pairs of opposite sides are parallel, that it has four right angles, and that not all four sides are congruent. Let's find the slopes of the sides using the Slope Formula.
| Side | Slope Formula | Simplified |
|---|---|---|
| Slope of XY: ( 1,- 2), ( 3,- 6) | - 6-( -2)/3- 1 | - 2 |
| Slope of YZ: ( 3,- 6), ( - 5, - 10) | - 10-( - 6)/- 5- 3 | 1/2 |
| Slope of ZW: ( - 5, - 10), ( - 7, - 6) | - 6-( - 10)/- 7-( -5) | - 2 |
| Slope of WX: ( - 7, - 6), ( 1,- 2) | - 2-( - 6)/1-( - 7) | 1/2 |
We can tell that the slopes of the opposite sides of our quadrilateral are equal. Therefore, both pairs of opposite sides are parallel and the quadrilateral is a parallelogram. Additionally, the consecutive side are perpendicular, as their slopes are opposite reciprocals. - 2 * 1/2 = -1 Therefore, our parallelogram is either a rectangle or a square. Finally, we need to check the lengths of the sides. To do this we will use the Distance Formula.
| Side | Distance Formula | Simplified |
|---|---|---|
| Length of XY: ( 1,- 2), ( 3,- 6) | sqrt(( 3- 1)^2+( - 6-( - 2))^2) | sqrt(20) |
| Length of YZ: ( 3,- 6), ( - 5, - 10) | sqrt(( - 5- 3)^2+( - 10-( - 6))^2) | sqrt(80) |
| Length of ZW: ( - 5, - 10), ( - 7, - 6) | sqrt(( - 7-( - 5))^2+( - 6-( - 10))^2) | sqrt(20) |
| Length of WX: ( - 7, - 6), ( 1,- 2) | sqrt(( 1-( - 7))^2+( - 2-( - 6))^2) | sqrt(80) |
As we can see, our quadrilateral does not have all sides of equal length, so it cannot be a square. Therefore, we have proved that it is a rectangle.